The Lie derivation of multivector fields along multivector fields has been introduced by Schouten (see cite{Sc, S}), and studdied for example in cite{M} and cite{I}. In the present paper we define the Lie derivation of differential forms along multivector fields, and we extend this concept to covariant derivation on tangent bundles and vector bundles, and find natural relations between them and...

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds (M, g) for the following general heat equation ut = ∆V u+ au log u+ bu where a is a constant and b is a differentiable function defined onM×[0,∞). We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimat...

Let M be a compact spin manifold with a smooth action of the ntorus. Connes and Landi constructed θ-deformations Mθ of M , parameterized by n × n skew-symmetric matrices θ. The Mθ’s together with the canonical Dirac operator (D,H) on M are an isospectral deformation of M . The Dirac operator D defines a Lipschitz seminorm on C(Mθ), which defines a metric on the state space of C(Mθ). We show tha...

For a large class of admissible functions f : R 7→ C, the operator derivatives dj dxj f(H0 + xV ), where H0 and V are self-adjoint operators on a separable Hilbert space H, exist and can be represented as multiple operator integrals [1, 14]. Let M be a semi-finite von Neumann algebra acting on H and τ a semi-finite normal faithful trace on M. For H0 = H ∗ 0 affiliated with M and V = V ∗ in the ...

 For a Banach algebra $fA$, we introduce ~$c.c(fA)$, the set of all $phiin fA^*$ such that $theta_phi:fAto  fA^*$ is a completely continuous operator, where $theta_phi$ is defined by $theta_phi(a)=acdotphi$~~ for all $ain fA$. We call $fA$, a completely continuous Banach algebra if $c.c(fA)=fA^*$. We give some examples of completely continuous Banach algebras and a sufficient condition for an o...

We also discuss the relation of our work to the earlier work on the decomposition of the ζ-determinant by Burghelea, Friedlander and Kappeler (from this point on referred to as BFK). The present work is companion to the paper [10] and in several places we refer to [10] for the proof of a given statement and a more detailed discussion. Let D : C(M ;S) → C(M ;S) be a compatible Dirac operator act...

For a von Neumann algebra M acting on a Hilbert space H with a cyclic and separating vector ξ0, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (M, ξ0). For a general Lindblad type generator L of a conservative quantum dynamical semigroup on M, we give sufficient conditions so that the operator H induced by L via the symmetric embedding of M...

we characterize compact composition operators on real banach spaces of complex-valued bounded lipschitz functions on metric spaces, not necessarily compact, with lipschitz involutions and determine their spectra.

Nonlocal operator basis from the path representation of the M(k + 1, k + 2) and the M(k + 1, 2k + 3) minimal models ABSTRACT We reinterpret a path describing a state in an irreducible module of the unitary minimal model M(k + 1, k + 2) in terms of a string of charged operators acting on the module's ground-state path. Each such operator acts non-locally on a path. The path characteristics are t...

Let X be a real separable Banach space and X * be its dual space. Let B(X) be Borel σ-field of X. For finite dimensional subspace F of X * we define the cylinder set C based on F as follows C = {x ∈ X; It is easy to show that C(X, X *) is a fileld. Let C(X, X *) be the σ− field generated by C(X, X *). Then C(X, X *) = B(X). If µ is a probability measure on (X, B(X)) satisfying ∫ X ∥x∥ 2 dµ(x) <...